Part of going from a state on the doubled Hilbert space to desntity matrices on it, has to do with averaging over microscopic information (keeping macroscopic data of the wormhole, such as its length, fixed). An important clue for how to do so is the following. After averaging over couplings, the SYK model has an \(O(N)\)  symmetry under rotation of the basis of fermions, \(\psi _i = O_i^j \psi _j\) . When we consider two copies of the SYK model, because we have a single average over the couplings, the averaged system is invariant under a diagonal \(O(N)\)  symmetry which rotates both the left and right fermions simultaneously. There is a similar symmetry in the random spin model. We will argue that this rotation is an example of the kind of microscopic details we should average over in constructing duals of bulk wormholes - two configurations that differ by an \(O(N)\)  rotation will look the same gravitationally. Thus we will consider \(O(N)\) -invariant density matrices \(\rho \) , or more precisely, density matrices where the only terms which break the \(O(N)\)  symmetry are explicit insertions of the Hamiltonian.
In section  we set up the specific class of density matrices we consider, and in section  we use the chord diagram approach to calculate two-sided correlation functions in this class of density matrices. We show that in a suitable limit the results obtained from our microscopic calculation reproduce the holographic calculation of correlations in a shockwave perturbation of TFD in [1]}.
